You mentioned Easter Eggs by Robert Spencer Carr. The description doesn't seem to match, but here it is:
Alien visitation as a chance to espouse American moral superiority vis-a-vis the USSR: one Martian lands in Washington D. C. (in a ship conveniently marked with the astronomical symbol for Mars) and another lands outside the Kremlin. Martians, we learn, are an ancient species who are dying out due to lack of the four precious elements which make Earth the “Garden of Eden of our solar system:” oxygen, water, earth, and blue sky (?). Even though they apparently require the same environment as humans, we are forcefully reminded nonetheless that theirs is an “alien intelligence, aloof, inhuman, inconceivably remote from our ways of reasoning.” Despite this, each is swayed by the arguments of their respective hosts – with the American offering a “fifth essential element of life… Its name is freedom” – and they become a pair of Cold Warriors and use their ships to fight to the death. The victor returns to Mars to summon forth a host of his species in order to enforce the victory of his side – but no one on Earth is sure which was the winner.
Also featuring a buffoonish general who refuses to listen to reason or scientists (“No life on Mars. Subject is closed.” To which an Einstein-ish character says “In science, no subject is ever closed. Only minds.”) and a woman who at first refuses to give up her fulfilling secretarial job in the White House to move to the Midwest with her fiancé, only to have the events of the story remove her “confusion.”
It was also released as Those Men from Mars and The Invaders.
It appeared in at least two different collections. Beyond Infinity of works by Carr and The Best Science-Fiction Stories: 1950 from a number of authors, edited by Everett F. Bleiler and T. E. Dikty. The latter also appeared as the Science Fiction Omnibus which added the stories from The Best Science-Fiction Stories: 1949.
As I said, this doesn't seem to match your description. I'm including it in case they just left out the parts you remember, or if you are remembering a different story from one of the same collections.